Electro-osmotic properties of porous permeable films

Permeable porous coatings on a flat solid support significantly impact its electrostatic and electrokinetic properties. Existing work has employed the mean-field Poisson-Boltzmann theory by focusing on simplified cases, such as weakly charged and/or thick porous films compared to the extension of an electrostatic diffuse layer. In this paper, we obtain a closed-form analytical solution for electrostatic potential profiles by lifting the assumptions of both a small volume charge density and a thick film. Our analysis provides a framework for interpreting and predicting superproperties specific to porous films of an arbitrary thickness, from an enhanced ion absorption to a consequent amplification of electro-osmotic flows due to emergence of slip velocity at an interface with an outer electrolyte leading to a large zeta potential. The latter can be tuned by varying the amount of added salt and remains finite at even high concentrations. Our theory is valid for systems obeying the nonlinear Poisson-Boltzmann equation and the results are relevant for hydrogel coatings, porous carbon and silica, polyelectrolyte brushes, and more.

Figure 1

Porous film of thickness $H$ in contact with an electrolyte solution. Anions and cations are denoted with bright and dark circles. An outer electrostatic diffuse layer of a thickness of the order of ${\lambda }_D\equiv {\kappa }^{-1}$ is formed in the vicinity of the coating. A tangential electric field, $E$, leads to a solvent flow of velocity $v$ (shown by right arrows).

Figure 2

A distribution of a potential built up by a film of $\rho =10$, calculated numerically for $\kappa H=0.3$ (dashed curve) and 3 (solid curve). Filled circles correspond to calculations from Eqs. (7) and (14), when $z/H\le 1$, and from Eq. (2), when $z/H\ge 1$. Open squares are obtained from Eq. (15).

Figure 3

Potentials at wall (solid lines) and surface (dashed) as a function of $\kappa H$ computed for fixed $\rho =20$ (upper set of curves) and $\rho =2$ (lower curves). Filled squares illustrate calculations from Eq. (6), circles are then obtained using Eq. (3). Open squares show results obtained using Eq. (9). Dash-dotted line is calculated from Eq. (10).

Figure 4

Potentials at wall (solid lines) and surface (dashed) as a function of $\rho$ computed for fixed $\kappa H=3$ (upper set of curves) and $\kappa H=0.2$ (lower curves). Filled squares illustrate calculations from Eq. (6), circles are then obtained using Eq. (3). Open squares show predictions of Eq. (9).

Figure 5

Ion concentration profiles computed at $\rho =10$ using $\kappa H=0.2$ (dashed curves) and 5 (solid curves). Local concentrations of anions (filled circles) and cations (open circles) are calculated using Eq. (2) when $z/H\ge 1$, and from Eqs. (7) and (14) when $z/H\le 1$.

Figure 6

Ion-enrichment at the wall, $c_0/2c_{\infty }=cosh{\psi }_0$, vs $\rho$ computed using $\kappa H=3$ (upper curve) and 0.3 (lower curve). Open and filled circles indicate results obtained using ${\psi }_0$ calculated from Eqs. (6) and (11).

Figure 7

(a) Electroosmotic velocity profiles computed using $\kappa H=1, \rho =5$, and $\mathcal{K}H=0$, 2, and $\infty$ (solid curves from top to bottom). Filled and open symbols indicate calculations from Eqs. (17) and (19). Squares are obtained using $v_s=0$, circles correspond to $v_s={\psi }_s+v_{\infty }$, where $v_{\infty }$ is given by Eq. (20); (b) computed upper bounds ($\mathcal{K}H\to 0$) on $v_{\infty }$ vs $\kappa H$ (solid curves). From top to bottom $\rho =20,10,$ and 2. Circles show results of calculations from Eq. (20).

Figure 8

$\zeta$ as a function of $c_{\infty }$ computed for a film of $H=25$ nm, $\varrho =150$ kC/${\mathrm{m}}^3$, using $\mathcal{K}H=0$, 1.5, and $\infty$ (solid curves from top to bottom). Dashed curve shows $|v_s|$ at $\mathcal{K}H=0$. Squares, open and filled circles plot results of calculations from Eqs. (21, 22, 23).